This is actually something Scott Bradfield used to make me do. It was one of my first exercises, just to improve my awareness. He'd take a manuscript, slaughter the first page, and then we'd slaughter the rest together. Within a month my prose was clearer and my stories were shorter. Prose, I came to understand, is a conduit from reader to content. If something's in the way, return with a battering ram.
Take, as an extreme example, a novel where we're following the POV of a loquacious struggling painter, who ineptly attempts to capture the beauty around him, which is his one true obsession. Let's say it's written in the form of a diary. Let's say that in the first chapter, you've done your job. The reader is in love with this character and his voice. He's loquacious. Strings of unnecessary pronouns and "ad-words" are part of his voice. This might be a harder story to make work, at least on the level of prose, but it's a telling example. In understanding a guideline you must always be aware of the exceptions. If you aren't, you don't understand the guideline.
"Kill the darlings" is good for students to hear, because the craft of writing is all about acquiring empathy with the reader. If you're trying to show your flare with words, you're focusing in the wrong direction. Beautiful writing, from the poetic to the stark, springs from empathy.
Note, "Kill the darlings" also refers to scene choices, and the same concept applies. Don't try to show how cool a scene is. Think of what the story needs from it. Then amplify, and it'll be a darling worth keeping.